Can all graphs be expressed as a function i.e. $f(x)=y$?

Question

Is there a function for every possible graph? If not, why not? Let it be continuous and with domain of all real numbers.

Answer 1

Let's say I draw a random curve (assuming it can be infinitely thin), can I find a proper function for it? Let the curve be continuous and with its domain of all real numbers.

There's two cases here. After you've drawn your curve (lets refer to your curve as $\Gamma_f$, for brevity's sake) in $\mathbb{R}^2$, attempt to find somewhere where can draw a vertical line that intersects your curve twice.

1. If you found a vertical line that intersects $\Gamma_f$ twice, there is no such function.
2. If you are unable to find such a vertical line, then there exists a function.

Let's elaborate now. First, note that a function is a "mapping" from one set to another (in this case, it is from $\mathbb R \to \mathbb R$). Functions that go from $A \to B$ have one crucial requirement: $$(\forall a \in A)(\exists ! b \in B) \qquad f(a)=b$$ in English, this translates to "for every element $a$ in $A$, there's only one $b$ in $B$ such that $f(a)=b$". That is, you can't have the element $a$ be mapped to two elements simultaneously, saying "$f(a)=b_1, b_2$" in this case is nonsensical. This is why if #1 was the case, then the curve $\Gamma_f$ does not correspond to a function.

Now let's look at the second case. Here, your curve $\Gamma_f$ corresponds to a function because it fits the requirements. Doesn't even need to be continuous, if you threw random dots all over the place then $\Gamma_f$ still corresponds to a function.

Now of course this function cannot be described with a polynomial if it is just a collection of points, you would have to describe it explicitly. For instance, the mapping $\mathbb R \to \mathbb R$ defined by $$f= \{ (1,1), \ (2, 5), \ (8,9) \}$$ is a function. We'd have $f(1)=1, f(2)=5,$ and $f(8)=9$. As for $f(25)$? That's undefined. Regardless, $f$ is still a function.

And that is why if #2 is your case, then your curve $\Gamma_f$ corresponds to a function.